Streamlining Calculus with u-Substitution: Simplifying the Integral of cos(2x)dx

Calculus is a powerful tool in the mathematics arsenal, and understanding various integration techniques is crucial for problem-solving. One such technique is u-substitution, which simplifies complex integrals by transforming them into more manageable forms. In this article, we will explore how u-substitution can be used to solve integrals involving trigonometric functions, specifically cos(2x)dx. We will walk through a comprehensive example to make the process clear and demonstrate its utility.

The Basics of u-Substitution

u-substitution is a method of integration that is used to simplify integrals by changing the variable of integration to a new variable, u. The method involves the following steps:

Choose a suitable expression, u, that is a function of the variable of integration. Compute the differential du in terms of the original variable of integration. Rewrite the integrand in terms of u and du. Perform the integration with respect to u. Substitute back the original variable of integration if necessary.

Applying u-Substitution to Integral of cos(2x)dx

Let's start by solving the integral ∫cos(2x)dx. We will use the u-substitution method to simplify this integral.

Step 1: Choose the Substitution

The first step is to choose a suitable substitution. In this case, we can let:

ucos(2x)

Step 2: Compute the Differential

We need to compute the differential du in terms of dx. Differentiating both sides of the equation with respect to x, we get:

dudx-2sin(2x)

Therefore, we can write:

du-2sin(2x)dx

Step 3: Rewrite the Integrand

Now, we rewrite the integrand in terms of u and du. Notice that:

cos(2x)dxdu-2sin(2x)

Since we know that:

du-2sin(2x)dx

We can substitute and simplify the integrand to:

cos(2x)dxdududu

Thus, the integral becomes:

∫cos(2x)dx-12∫1u·du

Step 4: Perform the Integration

We now integrate with respect to u:

-12∫1udu-12ln|u| C

Step 5: Substitute Back the Original Variable

To get the final answer in terms of x, we substitute ucos(2x) back into the equation:

-12ln|cos(2x)| C

This is the solution to the integral of cos(2x)dx using u-substitution.

Conclusion

Through this example, we have demonstrated how u-substitution simplifies the process of integration, particularly for integrals involving trigonometric functions. By choosing an appropriate substitution, performing the necessary steps, and substituting back the original variable, we can solve integrals that would otherwise be more complex and challenging. Understanding and applying u-substitution are valuable skills in calculus, enhancing both mathematical comprehension and problem-solving abilities.

Further Reading and Resources

For a deeper understanding of u-substitution and its applications in calculus, you may wish to explore the following resources:

Integration by Substitution - University of British Columbia U-substitution in Calculus II - Lamar University Khan Academy - U-Substitution

Using these resources, you can further enhance your knowledge and understanding of u-substitution and its applications in calculus.